Mathematics

Problem

Consider an economy with an aggregate production function given by

Y = 9K^{1/3}L^{2/3}

, with

Y

output,

K

the capital stock and

L

the labour force. Competitive factor markets guarantee that capital is paid its marginal product

MPK

. a. Show that

MPK

can be written as a function of capital per worker (denoted

k

). b. Show that the average product of capital

APK

(computed as

Y/K

) can also be written as a function of

k

. c. Which one is greater,

MPK

APK

? Explain the economic intuition of this. d. If the capital-output ratio were to remain constant over time, what would happen to

MPK

and

APK

in this economy? e. If the capital-output ratio were to remain constant over time, what would happen to the standard of living? Explain your answer.